Acylindrical arc choices and connected covering spaces in Charitos's new proof of the virtual Haken theorem

A 2025 preprint of Charitos (arXiv:2510.00617) proposes a new, purely geometric-topological proof of the virtual Haken conjecture, a statement that has been a theorem since Agol's 2013 proof via special cube complexes. The preprint constructs, for every compact, orientable, irreducible, atoroidal 3-manifold with infinite fundamental group, a decomposition as a handlebody glued to a manifold $N$ obtained from the exterior of a specially chosen surgery link in the 3-sphere by removing neighborhoods of arcs joining consecutive link components. The paper explicitly leaves two questions open: (1) whether these joining arcs can always be chosen so that $N$ is acylindrical, which by Thurston's hyperbolization theorem would give $N$ a complete hyperbolic structure with totally geodesic boundary (Remark 17); and (2) whether the final compression-and-lifting argument, which requires the constructed finite cover of $N$ to be connected, extends to covers with several components, and whether a connected cover can always be arranged (Remark 25).

Background

A closed 3-manifold is Haken if it contains a two-sided embedded closed surface that is incompressible, meaning the inclusion induces an injection on fundamental groups; it is irreducible if every embedded 2-sphere bounds a 3-ball. The virtual Haken conjecture predicts that every compact, orientable, irreducible 3-manifold with infinite fundamental group has a finite cover that is Haken. Attributed to Waldhausen and popularized through the standard problem lists of the era, it survived the proof of geometrization (Thurston's decomposition of 3-manifolds into geometric pieces, completed by Perelman) precisely in the hyperbolic case and was settled by Agol, with an appendix by Groves and Manning, using the machinery of special cube complexes — spaces glued from cubes along faces whose hyperplane structure is controlled well enough that the fundamental group virtually embeds in a right-angled Artin group; the statement is therefore a theorem. Charitos proposes an alternative, direct proof in the spirit of Thurston's program, and it is this new proof whose two explicitly flagged gaps form the present problem.

Two of Thurston's hyperbolization theorems frame the first gap. A compact, orientable, irreducible 3-manifold that is atoroidal (admits no essential, that is incompressible and non-boundary-parallel, embedded torus), has nonempty incompressible boundary, and contains no essential properly embedded annulus — a condition summarized by calling the manifold acylindrical, where an essential annulus is an incompressible properly embedded annulus not parallel into the boundary — admits a hyperbolic structure with totally geodesic boundary; if the annulus condition is dropped, the interior still admits a complete hyperbolic structure. The geodesic-boundary version is strictly stronger information about the piece.

Charitos's construction runs as follows. By classical surgery theory as recorded in the paper, every closed, orientable 3-manifold $M$ is obtained by Dehn surgery — removing the interior of a tubular solid-torus neighborhood of each link component and regluing a solid torus so that the slope specified by the surgery coefficient bounds a disc in the reglued torus — with coefficients $\pm1$ on a link $L=L_1\cup\cdots\cup L_n$ in the 3-sphere $S^3$ whose components are unknotted and whose pairwise linking numbers (the integers measuring how one oriented component winds around another) are $0$ or $\pm1$ ; a minimality condition on the number of components is imposed, and the paper derives an additional combinatorial property of the linking numbers (its property (v), stated explicitly in the problem statement below). Writing $N_0$ for the link exterior, the paper proves $N_0$ irreducible, atoroidal, acylindrical, and boundary-incompressible. One then joins consecutive components $L_i$ and $L_{i+1}$ by arcs $a_i$ and lets $H$ be a closed tubular neighborhood of the resulting graph; $H$ is a handlebody (a 3-ball with 1-handles attached, equivalently a regular neighborhood of a graph) of genus $n$ containing the surgery solid tori, and $N=\overline{S^3\setminus H}$ is proved, for arbitrary choices of the arcs, to be irreducible, atoroidal, and to have connected incompressible boundary of genus $n$ , with $M$ recovered by gluing a handlebody to $N$ along their common boundary (a handlebody filling). The final sections adapt the theorem of Cooper, Long, and Reid — every compact 3-manifold with incompressible boundary and hyperbolic interior either is covered by a product $F\times I$ or contains an essential immersed closed surface that lifts to an embedded incompressible surface in a finite cover — to produce an essential surface in a finite cover of $M$ . Within this program the paper records two unresolved points, its Remarks 17 and 25, which are the subject of this record; the author states that his main theorem does not depend on the first of them, and acknowledges the second as a limitation of the present argument.

Problem Statement

Let $M$ be a compact, orientable, irreducible, atoroidal 3-manifold with infinite fundamental group and let $L=L_1\cup\cdots\cup L_n\subset S^3$ be a surgery link produced by the construction of Charitos (arXiv:2510.00617): each $L_i$ is unknotted, all pairwise linking numbers lie in $\{0,\pm1\}$ , surgery coefficients are $\pm1$ , the number of components is minimal among links satisfying these conditions, and the link satisfies the paper's property (v): for every subfamily $\{L_{i_1},\dots,L_{i_k}\}\subseteq\{L_1,\dots,L_n\}$ there exist a member $L_{i_j}$ of the subfamily and a component $L_j\in\{L_1,\dots,L_n\}$ such that $\mathrm{lk}(L_j,L_{i_j})\neq0$ while $\mathrm{lk}(L_j,L_{i_m})=0$ for every other member $L_{i_m}$ of the subfamily, where $\mathrm{lk}$ denotes linking number. For arcs $a_1,\dots,a_{n-1}$ joining consecutive components, let $H$ be a closed tubular neighborhood of the graph $L\cup a_1\cup\cdots\cup a_{n-1}$ , a handlebody of genus $n$ , and let $N=\overline{S^3\setminus H}$ ; by Theorem 15 of the paper, $N$ is irreducible, atoroidal, and has incompressible boundary for any choice of the arcs, and $M$ is recovered by a handlebody filling of $N$ . Resolve the two questions the paper explicitly leaves open.

(Q1, Remark 17) Can the arcs $a_i$ always be chosen so that $N$ is acylindrical, that is, contains no essential properly embedded annulus? By Thurston's hyperbolization theorem for manifolds with boundary, this is equivalent to asking whether the arcs can be chosen so that $N$ admits a hyperbolic structure with totally geodesic boundary. The author states that the main theorem of the paper does not depend on this question.

(Q2, Remark 25) The proof of the virtual Haken theorem in the paper (Theorem 24) passes to a finite cover $\widetilde{N}\to N$ on which an essential surface $\widetilde{S}$ is compressed inside the filled cover $\widetilde{M}$ ; the argument requires $\widetilde{N}$ to be connected, since otherwise $\widetilde{M}$ is assembled from more than one handlebody and key steps fail — for instance, the existence of a curve of $\widetilde{S}$ bounding a disc in $\widetilde{N}$ , and the dichotomy placing all compression pieces inside or outside the 3-ball bounded by a sphere component, are no longer guaranteed. Under assumption (A) — that $\partial T$ does not separate $\partial N$ , where $T\subset N$ is the incompressible surface used to construct the cyclic cover, an assumption arranged by the double-covering construction that version 2 of the paper adds — the cyclic cover $\widetilde{N}$ is connected by construction, so what remains open is whether that connectedness is established in complete generality within the paper's framework and whether the argument can be freed of the hypothesis. The question has two independently refutable branches: (Q2a) whether the compression-and-lifting argument of Theorem 24 can be made to work when the cover $\widetilde{N}$ has several components, and (Q2b) whether the covering construction of the paper (including the double-covering step that arranges the boundary of the constructed surface to be non-separating) can always be arranged to yield a connected $\widetilde{N}$ , equivalently whether assumption (A) can be removed. Q2 is settled by an affirmative resolution of either branch, which frees the proof of the limitation, or by negative resolutions of both, which establishes the limitation as genuine.

Both questions are to be resolved, each affirmatively or negatively.

References

Charalampos Charitos, A new proof of the virtual Haken conjecture, arXiv:2510.00617 (v1, 1 Oct 2025; v2, 13 Nov 2025), DOI 10.48550/arXiv.2510.00617, https://arxiv.org/abs/2510.00617.
Ian Agol, with an appendix by Ian Agol, Daniel Groves, and Jason Manning, The virtual Haken conjecture, Documenta Mathematica 18 (2013), 1045-1087, DOI 10.4171/dm/421, arXiv:1204.2810, https://arxiv.org/abs/1204.2810.
D. Cooper, D. Long, and A. Reid, Essential closed surfaces in bounded 3-manifolds, Journal of the American Mathematical Society 10 (1997), 553-563, DOI 10.1090/S0894-0347-97-00236-1, https://doi.org/10.1090/S0894-0347-97-00236-1.
Luis Celso Chan Palomo and Scott A. Taylor, Hyperbolic Brunnian Theta Curves, arXiv:2512.14533 (2025), DOI 10.48550/arXiv.2512.14533, https://arxiv.org/abs/2512.14533.
Colin Adams, Francisco Gomez-Paz, Jiachen Kang, Lukas Krause, Gregory Li, Chloe Marple, and Ziwei Tan, Hyperbolic Handlebody Complements in 3-Manifolds, arXiv:2501.19330 (2025), DOI 10.48550/arXiv.2501.19330, https://arxiv.org/abs/2501.19330.
William P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bulletin of the American Mathematical Society 6 (1982), 357-382, DOI 10.1090/S0273-0979-1982-15003-0, https://doi.org/10.1090/S0273-0979-1982-15003-0.
Friedhelm Waldhausen, On irreducible 3-manifolds which are sufficiently large, Annals of Mathematics 87 (1968), 56-88, DOI 10.2307/1970594, https://doi.org/10.2307/1970594.